Stochastic fixed point equations for insurance and financial time series modeling

Specialeforsvar ved Martin Gennerskov Nielsen 

Titel: Stochastic fixed point equations for insurance and financial time series modeling

Abstract: The focus of this thesis is tail estimates for stochastic fixed point equations of the form V =^D f(V), where f(v) = (Av + B)^+. Tail estimates can be used to examine the probability of ruin in for instance the "Ruin problem with stochastic investments" and in financial time series models. We will include theoretical results such as Goldie's theorem stating P(V > u) ~ Cu^{-\xi} as u tends to infinity, and a result characterizing the constant of tail decay C. For the latter result, a dual change of measure will be introduced, to analyse the behaviour of V. Besides theoretical analysis of the tail estimates, simulation will be applied as well. As P(V > u) is a rare event, importance sampling will be applied in the simulation algorithm. The importance sampling will utilize the dual change of measure such that the recursion of the stochastic fixed point equation is generated in the \xi-shifted measure until exceeding level u, and then in the original measure, making a regeneration cycle. The inclusion of importance sampling ensures efficiency of the algorithm. 

Vejleder:    Jeffrey F. Collamore
Censor:       Mette Havning